Plain-English statement
On the bundled generator domain, the right derivative of the semigroup orbit at time t is P_t applied to the generator.
Mathematical statement
For f in D(L), [P_(t+h)f - P_t f]/h tends to P_t(Lf) as h decreases to zero through positive times.
Intuition
The semigroup law turns the orbit increment at t into P_t applied to the generator quotient at zero.
Conditions
- a continuous-linear semigroup on a real normed space
- membership of f in its right-generator domain
Why these conditions cannot be dropped
- domain membership supplies the generator limit
- bounded linearity of P_t transports that limit
- this is a backward right-derivative theorem, not a forward Fokker-Planck equation
Proof route
- obtain the canonical right-generator limit of f
- apply the existing unbundled backward-equation equivalence
- select its Tendsto direction
Lean interface notes
- nhdsWithin 0 (Ioi 0) records the one-sided positive-time limit
- the concrete Langevin generator and its closed core remain outside this theorem
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Lean statement
theorem kolmogorov_backward_right_generator
(S : ContinuousLinearSemigroup M)
(f : generatorDomainSubmodule S) (t : ℝ≥0) :
Tendsto
(fun h : ℝ≥0 => rightOrbitDifferenceQuotient S t h (f : M))
(nhdsWithin 0 (Ioi 0))
(𝓝 (S.op t (rightGenerator S f))) := by
exact (kolmogorov_backward_right S (rightGeneratorValue_spec S f) t).2
end
end OperatorGeneratorDomain
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheoryOpen AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:203published source at 7bcd37294df1